Quarterback Andrew Luck led the fightback by engineering five second-half touchdown drives, capping the comeback with a 64-yard pass to T.Y. Hilton for the go-ahead score with 4:21 left on the clock in Indianapolis on Saturday.
"We never panicked," said Luck of the second-largest comeback in NFL playoff history. "We feel like we are in every game. The guys stuck with me and we pulled this one out."
The Colts stumbled out the gate as Luck threw three interceptions that allowed the Chiefs to take a 38-10 lead.
Luck, who is in his second year in the league, finished with four touchdown passes, recovered a fumble for another touchdown and completed 29 of 45 passes for a total of 443 yards.
The victory means the Colts could potentially face former franchise hero Peyton Manning, who now plays quarterback for Denver. That matchup will depend on the result of the Cincinnati and San Diego wild card game on Sunday (Monday AEDT).
If Cincinnati wins, then Denver will host the Colts on January 11. If the Chargers win, the Colts will square off against New England on January 12.
TY Hilton also stepped up against the Chiefs, making 13 catches for 224 yards and a pair of touchdowns. Donald Brown had a receiving and rushing touchdown for Indianapolis, who had dropped three straight games in the playoffs after last season's 24-9 wild-card loss to the eventual Super Bowl champion Baltimore Ravens.
Kansas City got off to a rocky start when all-star running back Jamaal Charles left the game after suffering a concussion on the opening drive of the game.
Charles led the AFC in rushing for the second consecutive season, had an NFL-best 19 total touchdowns, and averaged 132 rushing yards per game in 2013.
The Chiefs shook off the loss of Charles to score on their first five possessions and build a substantial lead.
Quarterback Alex Smith threw four touchdown passes while going 30-of-46 for 378 yards in the loss.
Dwayne Bowe made eight catches for 150 yards and a touchdown as the Chiefs suffered their eighth consecutive playoff loss since Hall-of-Famer Joe Montana led the franchise to a divisional round win during the 1993 season.
The AFC South champion Colts went into the game on a roll, having won three straight in which they allowed a total of 20 points. That span included an 23-7 win at Kansas City on December 22.
Taking the field later Saturday are the New Orleans Saints and Philadelphia Eagles in an NFC contest.
In the other NFC wildcard game of the weekend, San Francisco travels to Green Bay to play the Packers on Sunday.